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arxiv: 2512.18532 · v3 · submitted 2025-12-20 · ❄️ cond-mat.str-el · hep-th· quant-ph

Global approximations to correlation functions of strongly interacting quantum field theories

Yuanran Zhu , Yang Yu , Efekan K\"okc\"u , Emanuel Gull , Chao Yang This is my paper

Pith reviewed 2026-05-16 20:12 UTC · model grok-4.3

classification ❄️ cond-mat.str-el hep-thquant-ph
keywords correlation functionsPadé approximationstrong coupling expansionweak coupling expansionφ⁴ theoryHubbard modelMatsubara Green's functionanalytic continuation
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The pith

Two-point Padé approximants interpolate weak and strong coupling expansions to produce global approximations to correlation functions in strongly interacting quantum field theories.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a method that starts from known perturbative expansions at weak and strong coupling and uses two-point Padé interpolation to build approximations that cover the entire coupling range for correlation functions. Benchmarks on the lattice φ⁴ theory show that these approximants converge uniformly and globally to the exact result. On the two-dimensional Hubbard model the same construction, even when limited to second order, already reproduces the Matsubara Green's function reasonably well over a broad set of parameters. The approach therefore offers a practical route to non-perturbative information without performing a full non-perturbative calculation. A heuristic argument drawn from analytic function theory is given to account for the observed convergence behavior.

Core claim

By constructing two-point Padé approximants that match both the weak-coupling and strong-coupling perturbative series of a correlation function, one obtains a global approximation that, for the lattice φ⁴ theory, converges uniformly to the exact function and, for the two-dimensional Hubbard model, already furnishes a useful characterization of the Matsubara Green's function at second order across a wide parameter window.

What carries the argument

The two-point Padé expansion, which simultaneously matches the perturbative series at both weak and strong coupling and thereby interpolates the correlation function across the full coupling range.

If this is right

  • In the φ⁴ theory the two-point Padé approximants converge uniformly to the exact correlation function over the entire coupling range.
  • In the Hubbard model even a second-order Padé approximant already yields a reasonable description of the Matsubara Green's function for a wide range of interaction strengths and fillings.
  • The same interpolation procedure can be applied to other models once their weak- and strong-coupling perturbative expansions for the correlation function are known.
  • The convergence is supported by a heuristic argument based on the analytic properties of the correlation functions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The method could be extended to higher-point correlation functions or to real-frequency spectral functions by combining the Padé construction with analytic continuation techniques.
  • If the same uniform convergence holds for other bosonic or fermionic field theories, the approach would reduce the computational cost of mapping out phase diagrams that straddle weak and strong coupling.
  • One could test whether increasing the order of the input expansions systematically improves the approximation at intermediate couplings where neither perturbative series is reliable.

Load-bearing premise

The correlation functions admit analytic continuations whose properties permit the two-point Padé approximant to converge uniformly.

What would settle it

Compute the exact correlation function of the lattice φ⁴ theory at an intermediate coupling value and check whether the two-point Padé approximant built from low-order weak and strong expansions deviates from it by more than a few percent.

Figures

Figures reproduced from arXiv: 2512.18532 by Chao Yang, Efekan K\"okc\"u, Emanuel Gull, Yang Yu, Yuanran Zhu.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Schematic illustration of the convergence mechanism [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 1
Figure 1. Figure 1: Analytic continuation (AC) of the correlation functions for exactly solvable models. In each panel, [PITH_FULL_IMAGE:figures/full_fig_p015_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Off-diagonal correlation functions G0j (˜g) of the one-dimensional lattice ϕ 4 model. In each panel, the solid lines represent the ground-truth results obtained from Langevin Monte Carlo simulations, while the dashed lines show the corresponding approximations from the two-point Padé scheme (Padé-w6-s3). g. As a result, the two-point Padé expansion does not interpolate between two analytic germs as display… view at source ↗
Figure 3
Figure 3. Figure 3: (Left) Approximation of the imaginary part of the Hubbard dimer Green’s function, Im[G00(U, iωn)], using the Padé–Taylor scheme (Padé-w4–s3). The curves show interpolation results for all Matsubara frequencies iωn (Right) The corresponding Green’s function G00(U, τ ) in the imaginary-time domain. In both figures, solid lines represent the analytic solution and the dashed lines correspond to the Padé approx… view at source ↗
Figure 4
Figure 4. Figure 4: Approximation of the Matsubara Green’s function for the 2D Hubbard model with different [PITH_FULL_IMAGE:figures/full_fig_p017_4.png] view at source ↗
read the original abstract

We introduce a method for constructing global approximations to correlation functions of strongly interacting quantum field theories, starting from perturbative results. The key idea is to employ interpolation method, such as the two-point Pad\'e expansion, to interpolate the weak and strong coupling expansions of correlation function. We benchmark this many-body interpolation approach on two prototypical models: the lattice $\phi^4$ field theory and the 2D Hubbard model. For the $\phi^4$ theory, the resulting two point Pad\'e approximants exhibit uniform and global convergence to the exact correlation function. For the Hubbard model, we show that even at second order, the Pad\'e appproximant already provides reasonable characterization of the Matsubara Green's function for a wide range of parameters. Finally, we offer a heuristic explanation for these convergence properties based on analytic function theory.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces a method for constructing global approximations to correlation functions in strongly interacting quantum field theories by interpolating weak- and strong-coupling perturbative expansions via two-point Padé approximants. It benchmarks the approach on the lattice φ⁴ theory, claiming uniform and global convergence to the exact correlation function, and on the 2D Hubbard model, where second-order approximants already yield reasonable characterizations of the Matsubara Green's function over a wide parameter range. A heuristic explanation based on analytic function theory is offered to justify the convergence properties.

Significance. If the convergence claims hold beyond the specific models studied, the interpolation method could provide a useful tool for obtaining non-perturbative correlation functions from perturbative data alone, with potential applications in condensed-matter and quantum-field-theory calculations. The numerical benchmarks on φ⁴ and Hubbard models show promising results that merit further investigation.

major comments (2)
  1. [concluding heuristic explanation] The central claim of uniform and global convergence for the φ⁴ two-point Padé approximants (abstract and concluding section) rests solely on a heuristic argument from analytic function theory. No theorem is derived or cited guaranteeing that the relevant correlation functions are meromorphic (or possess isolated singularities) in a domain containing the interpolation path in the complex coupling plane; without singularity analysis or radius-of-convergence control, standard Padé theory indicates that uniform convergence on compact sets can fail in the presence of natural boundaries or dense singularities.
  2. [Hubbard-model results] For the Hubbard-model benchmarks (abstract and results section), the manuscript states that even second-order Padé approximants provide reasonable characterization across a wide parameter range, yet supplies no quantitative error metrics, implementation details, or systematic comparison against exact or higher-order references. This omission makes it difficult to assess whether the observed agreement supports the broader applicability claim or is limited to the specific regimes tested.
minor comments (2)
  1. Typo in the abstract: 'appproximant' should read 'approximant'.
  2. The notation for the two-point Padé expansion would benefit from an explicit formula or algorithmic description early in the methods section to aid reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive major comments. We address each point below and have prepared revisions to clarify claims and strengthen the presentation of results.

read point-by-point responses

  1. Referee: The central claim of uniform and global convergence for the φ⁴ two-point Padé approximants (abstract and concluding section) rests solely on a heuristic argument from analytic function theory. No theorem is derived or cited guaranteeing that the relevant correlation functions are meromorphic (or possess isolated singularities) in a domain containing the interpolation path in the complex coupling plane; without singularity analysis or radius-of-convergence control, standard Padé theory indicates that uniform convergence on compact sets can fail in the presence of natural boundaries or dense singularities.

    Authors: We agree that the convergence statement relies on numerical benchmarks in the lattice φ⁴ theory together with a heuristic argument rather than a general theorem. No rigorous proof of meromorphicity or isolated singularities for arbitrary correlation functions is provided, as such an analysis lies outside the scope of the present work. In the revised manuscript we will modify the abstract and concluding section to state that the two-point Padé approximants exhibit convergence to the exact results in the φ⁴ benchmarks, supported by the heuristic explanation, while explicitly noting the absence of a general singularity theorem and the possibility of natural boundaries in other models. revision: yes

  2. Referee: For the Hubbard-model benchmarks (abstract and results section), the manuscript states that even second-order Padé approximants provide reasonable characterization across a wide parameter range, yet supplies no quantitative error metrics, implementation details, or systematic comparison against exact or higher-order references. This omission makes it difficult to assess whether the observed agreement supports the broader applicability claim or is limited to the specific regimes tested.

    Authors: We accept that quantitative error measures and additional implementation details are needed for a clearer assessment. In the revision we will insert error tables and plots (e.g., relative L² deviations from reference data) for the Matsubara Green’s function across the parameter ranges shown, supply the precise two-point Padé construction algorithm used, and include direct comparisons with higher-order perturbative results where available. revision: yes

Circularity Check

0 steps flagged

No circularity: standard two-point Padé interpolation applied to independent perturbative series; convergence supported by benchmarks and heuristic, not by construction.

full rationale

The derivation begins with externally computed weak-coupling and strong-coupling perturbative expansions of the correlation functions, then applies the standard two-point Padé interpolation procedure to produce the global approximant. Numerical benchmarks on the φ⁴ theory and Hubbard model are used to assess accuracy, while the convergence argument is presented as a heuristic based on general analytic-function properties rather than a self-referential theorem or fitted parameter. No equation reduces the approximant to its inputs by definition, no prediction is statistically forced from a subset of the same data, and no load-bearing step relies on self-citation chains. The method is therefore self-contained against external perturbative inputs and direct numerical checks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The approach assumes the existence of both weak and strong coupling expansions and the applicability of interpolation methods without introducing new parameters or entities.

axioms (1)
  • domain assumption The correlation functions are analytic in the complex plane or have suitable analytic properties allowing Padé convergence.
    Heuristic explanation based on analytic function theory mentioned in abstract.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Strong-coupling expansion and two-point Pad\'e approximation for lattice $\phi^4$ field theory

    hep-lat 2026-04 unverdicted novelty 7.0

    Two-point Padé approximants that interpolate weak- and strong-coupling expansions produce accurate global approximations to the two-point correlation function in lattice φ⁴ theory over wide coupling ranges.

Reference graph

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